We may assume \(\Vert \Omega \Vert _{L^{\infty }(\mathbb {S}^{d-1})}=1\) .
Obviously, it suffices to prove the estimate
\(|\Lambda ^{A}(f_1,\,f_2)|\lesssim r^{\prime }\sup _{\mathcal {S}}\mathcal {A}_{\mathcal {S},\,L\log L,\,L^r}(f_1,\,f_2)\) 

holds for all bounded functions \(f_1\)  and \(f_2\)  with compact supports.
To prove (REF ), we follow the main steps of Theorem C in [1]}. But we don't need to prove the self-adjoint property as in [1]}.
In addition, we only need to decompose \( f_1\) . These factors make our proof more elegant and clear. We divide the proof into two steps.
Step 1. Auxiliary estimate.
Using the Calderón-Zygmund decomposition
with respect to \(\mathcal {Q}\) . We may decompose \(f_1=b_1+g_1\) 
with \(b_1=\sum \limits _{\mathbb {L}\in \mathcal {Q}}(f_1-\langle f_1\rangle _\mathbb {L}){\chi }_\mathbb {L}\)  such that for any \(p\ge 1\) ,
\(\Vert g_1\Vert _{\mathcal {Y}_{\infty }(\mathcal {Q})}\lesssim \Vert f_1\Vert _{\mathcal {Y}_{p}(\mathcal {Q})}, \qquad \Vert b_1\Vert _{\dot{\mathcal {X}}_{p}(\mathcal {Q})}\lesssim \Vert f_1\Vert _{\mathcal {Y}_{p}(\mathcal {Q})}.\) 
Then
\(\Lambda ^A_{\mathcal {Q}}(f_1,\,f_2)= \Lambda ^A_{\mathcal {Q}}(g_1,\,f_2)+ \Lambda ^A_{\mathcal {Q}}(b_1,\,f_2).\) 

By inequality (REF ), it is obvious that
\(| \Lambda ^A_{\mathcal {Q}}(g_1,\,f_2)| \le r^{\prime }|Q|\Vert g_1\Vert _{\mathcal {Y}_{r^{\prime }}(\mathcal {Q})} \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}&\le r^{\prime } |Q|\Vert f_1\Vert _{\mathcal {Y}_{1}(\mathcal {Q})} \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}\\& \le r^{\prime }|Q| \Vert f_1\Vert _{\mathcal {Y}_{L\log L}(\mathcal {Q})} \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}.\) 

On the other hand, the fact that \( \Vert b_1\Vert _{\dot{\mathcal {X}}_{L\log L}(\mathcal {Q})}\lesssim \Vert f_1\Vert _{\mathcal {Y}_{L\log L}(\mathcal {Q})}\) , Lemma REF , together with Lemma REF  yield that
\(&|\Lambda ^A_{\mathcal {Q}}(b_1,\,f_2)|\le r^{\prime }|Q|\Vert b_1\Vert _{\dot{\mathcal {X}}_{L\log L}(\mathcal {Q})}\Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}\lesssim r^{\prime }|Q| \Vert f_1\Vert _{\mathcal {Y}_{L\log L}(\mathcal {Q})} \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})}.\) 

Therefore
\(| \Lambda _Q^A(f_1,f_2)|& \le |\Lambda _{\mathcal {Q}}^A(f_1,f_2)|+\sum \limits _{\mathbb {L}\in \mathcal {Q},\mathbb {L}\subset Q}|\Lambda _L^A(f_1,f_2)|\\&\le r^{\prime }|Q| \langle f_1\rangle _{{L\log L},3Q} \langle f_2\rangle _{{r},3Q} +\sum \limits _{\mathbb {L}\in \mathcal {Q},\mathbb {L}\subset Q}|\Lambda _\mathbb {L}^A(f_1\chi _\mathbb {L},f_2)|.\) 

Step 2. Iterative process.
Fix \(f_j\in L^{r_j}(\mathbb {R}^d)\) , \(j=1,2\)  with compact support, we choose the dyadic lattice \(\mathcal {D}\)  such that we can find a large enough dyadic cube \(Q_0\in \mathcal {D}\)  with supp \(f_1\in Q_0\) , supp \(f_2\in 3Q_0\) .
Set \(\mathcal {S}_0=\lbrace Q_0\rbrace \)  and let
\(\mathcal {S}_{k+1}\)  be the maximal cubes \(\mathbb {L}\in \mathcal {Q}\)  such that \(9 \mathbb {L}\subset \cup _{Q\in \mathcal {S}_k} E_{Q}\) 
\(E_Q:=\lbrace x\in 3Q: \max \lbrace  \frac{M_{L(\log L)}f_1 {\chi }_{3Q}(x)}{\Vert f_1\Vert _{L(\log L),\,3Q}} , \frac{M_{p}{(f_2 {\chi }_{3Q})}{(x)}}{{\langle f_2\rangle }_{r,3Q}}\rbrace >C_d\rbrace \)  for some
chosen constant \(C_d\) .
For all \(Q\in \mathcal {S}_0\) , one gets
\(|\Lambda ^{A}(f_1,\,f_2)|= | \Lambda _{Q_0}^A(f_1{\chi }_{Q_0},f_2{\chi }_{3Q_0})|\le r^{\prime }|Q_0| \Vert f_1\Vert _{\mathcal {Y}_{L\log L}(\mathcal {Q }_0)} \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q}_0)} +\sum \limits _{\mathbb {L}\in \mathcal {S}_{1}({Q_0}),\mathbb {L}\subset Q_0}|\Lambda _\mathbb {L}^A(f_1,f_2)|.\) 

For all \(k>0\)  and \(Q\in \mathcal {S}_k\) , we have
\(|\Lambda ^{A}(f_1,\,f_2)|\le r^{\prime }|Q| \Vert f_1\Vert _{\mathcal {Y}_{L\log L}(\mathcal {Q})} \Vert f_2\Vert _{\mathcal {Y}_{r}(\mathcal {Q})} +\sum \limits _{\mathbb {L}\in \mathcal {S}_{k+1}({Q}),\mathbb {L}\subset Q}|\Lambda _\mathbb {L}^A(f_1,f_2)|.\) 

When \(\inf \lbrace s_Q:Q\in \mathcal {S}_k\rbrace <s_{Q_0}\) , the iteration stops and there holds that
\(|\Lambda ^{A}(f_1,\,f_2)|\lesssim r^{\prime }\sum _{Q\in {\cup _{\kappa =0}^k \mathcal {S}_\kappa }}\langle |f|\rangle _{L\log L,\,3Q}\langle |g|\rangle _{3Q,\,r}|Q|.\) 
This finishes the proof of (REF ).
